IndisputableMonolith.Gravity.LatticeRicci

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Gravity.DiscreteCurvature
   4
   5/-!
   6# Step 4: Lattice Ricci Scalar Convergence (Linearized Case)
   7
   8Defines the lattice Ricci scalar from deficit angles and proves its
   9convergence to the continuum Ricci scalar in the linearized (weak-field)
  10regime.
  11
  12## The Key Result
  13
  14In linearized gravity with metric g = eta + h:
  15  R_continuum = -nabla^2(trace h) + div div h
  16
  17In harmonic gauge (div h_bar = 0):
  18  R = -nabla^2(trace h)
  19
  20The lattice Ricci scalar is defined as the sum of deficit angles
  21divided by dual areas. From DiscreteCurvature:
  22  curvature_from_deficit(d2h, a) = -d2h
  23
  24Summing over 3 axes:
  25  R_lattice = -(d2h_xx + d2h_yy + d2h_zz) = -nabla^2(trace h)
  26
  27This IS the continuum Ricci scalar (in harmonic gauge).
  28The convergence error is O(a^2) from the lattice Laplacian convergence.
  29
  30## Scope
  31
  32This module proves the LINEARIZED case only. The full nonlinear Regge
  33convergence (deficit angle -> Riemann tensor -> Ricci) requires
  34significantly more machinery and is stated as a hypothesis for the
  35nonlinear regime.
  36-/
  37
  38namespace IndisputableMonolith
  39namespace Gravity
  40namespace LatticeRicci
  41
  42open Constants DiscreteCurvature
  43
  44noncomputable section
  45
  46/-! ## Linearized Ricci Scalar -/
  47
  48/-- The linearized Ricci scalar from a metric perturbation h.
  49    In harmonic gauge: R = -nabla^2(trace h).
  50    We represent this via the 3 second derivatives of the trace. -/
  51noncomputable def continuum_ricci_linearized (d2h_xx d2h_yy d2h_zz : ℝ) : ℝ :=
  52  -(d2h_xx + d2h_yy + d2h_zz)
  53
  54/-- The lattice Ricci scalar: sum of deficit-angle curvatures over 3 axes. -/
  55noncomputable def lattice_ricci (d2h_xx d2h_yy d2h_zz a : ℝ) : ℝ :=
  56  curvature_from_deficit (deficit_angle_deformed d2h_xx a) (a ^ 2) +
  57  curvature_from_deficit (deficit_angle_deformed d2h_yy a) (a ^ 2) +
  58  curvature_from_deficit (deficit_angle_deformed d2h_zz a) (a ^ 2)
  59
  60/-! ## Convergence Theorem -/
  61
  62/-- **LINEARIZED RICCI CONVERGENCE THEOREM**:
  63    The lattice Ricci scalar equals the continuum linearized Ricci scalar
  64    EXACTLY (not just in the limit).
  65
  66    This is because each deficit-angle curvature = -d2h_kk exactly
  67    (from DiscreteCurvature.curvature_equals_d2h), and the sum gives
  68    -(d2h_xx + d2h_yy + d2h_zz) = R_continuum.
  69
  70    The O(a^2) error comes from the SAMPLING of h on the lattice
  71    (i.e., the finite-difference approximation of d2h), not from the
  72    deficit-angle-to-curvature step itself. -/
  73theorem lattice_ricci_equals_continuum (d2h_xx d2h_yy d2h_zz a : ℝ) (ha : a ≠ 0) :
  74    lattice_ricci d2h_xx d2h_yy d2h_zz a =
  75    continuum_ricci_linearized d2h_xx d2h_yy d2h_zz := by
  76  unfold lattice_ricci continuum_ricci_linearized
  77  rw [curvature_equals_d2h d2h_xx a ha,
  78      curvature_equals_d2h d2h_yy a ha,
  79      curvature_equals_d2h d2h_zz a ha]
  80  ring
  81
  82/-- The lattice Ricci scalar vanishes for flat space (all d2h = 0). -/
  83theorem lattice_ricci_flat (a : ℝ) (ha : a ≠ 0) :
  84    lattice_ricci 0 0 0 a = 0 := by
  85  rw [lattice_ricci_equals_continuum 0 0 0 a ha]
  86  unfold continuum_ricci_linearized; ring
  87
  88/-- The Ricci scalar is negative for positive Laplacian of trace h
  89    (e.g., a mass concentration that makes nabla^2 Phi > 0). -/
  90theorem ricci_negative_for_mass (d2h_xx d2h_yy d2h_zz : ℝ)
  91    (hpos : 0 < d2h_xx + d2h_yy + d2h_zz) :
  92    continuum_ricci_linearized d2h_xx d2h_yy d2h_zz < 0 := by
  93  unfold continuum_ricci_linearized; linarith
  94
  95/-! ## Connection to Poisson Equation -/
  96
  97/-- In linearized RS gravity, the Poisson equation is:
  98    nabla^2 Phi = (kappa/2) * rho
  99    where Phi is the Newtonian potential and rho is the defect density.
 100
 101    The Ricci scalar is R = -2 * nabla^2 Phi (for trace h = -2*Phi).
 102    So R = -kappa * rho.
 103
 104    This connects the lattice Ricci to the stress-energy source. -/
 105noncomputable def ricci_from_source (rho : ℝ) : ℝ :=
 106  -ZeroParameterGravity.kappa_rs * rho
 107
 108theorem ricci_source_proportional_to_kappa (rho : ℝ) :
 109    ricci_from_source rho = -(8 * phi ^ 5) * rho := by
 110  unfold ricci_from_source
 111  rw [ZeroParameterGravity.kappa_rs_closed_form]
 112
 113/-- Positive mass density gives negative Ricci scalar (attractive gravity). -/
 114theorem ricci_attractive (rho : ℝ) (hrho : 0 < rho) :
 115    ricci_from_source rho < 0 := by
 116  unfold ricci_from_source
 117  exact neg_neg_of_neg (mul_pos ZeroParameterGravity.kappa_pos hrho) |>.2
 118
 119/-! ## Nonlinear Regime (Hypothesis) -/
 120
 121/-- The FULL (nonlinear) Regge convergence: deficit angles on a refined
 122    lattice converge to the Riemann curvature tensor, and the summed
 123    Ricci scalar converges at O(a^2).
 124
 125    This is the content of Regge's 1961 theorem (confirmed numerically
 126    by Gentle-Miller 2001). We state it as a hypothesis for the
 127    nonlinear regime, with the linearized case proved above. -/
 128def H_nonlinear_regge_convergence : Prop :=
 129  ∀ (R_exact : ℝ) (a : ℝ), 0 < a → a < 1 →
 130    ∃ C : ℝ, 0 < C ∧
 131    ∀ d2h_xx d2h_yy d2h_zz : ℝ,
 132      |lattice_ricci d2h_xx d2h_yy d2h_zz a - R_exact| ≤ C * a ^ 2
 133
 134/-! ## Certificate -/
 135
 136structure LatticeRicciCert where
 137  linearized_exact : ∀ d2h_xx d2h_yy d2h_zz a : ℝ, a ≠ 0 →
 138    lattice_ricci d2h_xx d2h_yy d2h_zz a =
 139    continuum_ricci_linearized d2h_xx d2h_yy d2h_zz
 140  flat_vanishes : ∀ a : ℝ, a ≠ 0 → lattice_ricci 0 0 0 a = 0
 141  attractive : ∀ rho : ℝ, 0 < rho → ricci_from_source rho < 0
 142  kappa_coupling : ∀ rho : ℝ, ricci_from_source rho = -(8 * phi ^ 5) * rho
 143
 144theorem lattice_ricci_cert : LatticeRicciCert where
 145  linearized_exact := lattice_ricci_equals_continuum
 146  flat_vanishes := lattice_ricci_flat
 147  attractive := ricci_attractive
 148  kappa_coupling := ricci_source_proportional_to_kappa
 149
 150end
 151
 152end LatticeRicci
 153end Gravity
 154end IndisputableMonolith
 155